Lecture-07: the Thermodynamic Limit

Lecture-07: The Thermodynamic Limit

1 The thermodynamic limit

The main purpose of statistical physics is to understand the macroscopic behaviour of a large number, N 1, of microscopic components (atoms, molecules, etc.) under simple local interactions. For example, 23 in the case of water in a bottle, the number of H2O molecules N is typically of order 10 (18g of water contains approximately 6 × 1023 molecules), and this huge number leads physicists to focus on the N → ∞ limit, also called the thermodynamic limit.

1.1 The intensive thermodynamic potentials For large N, the thermodynamic potentials are proportional to N. The intensive thermodynamic potentials f (β), u(β), s(β) are defined as follows. Definition 1.1 (Intensive thermodynamic potentials). Denoting the thermodynamic potentials for N particle system as FN(β),UN(β),SN(β) for the free energy, the internal energy, and the canonical entropy re- spectively. We can define the free energy density, the energy density, and the entropy density as F (β) U (β) S (β) f (β) = lim N , u(β) = lim N , s(β) = lim N . (1) N→∞ N N→∞ N N→∞ N Partition function Z(β) is a sum of exponentials, and hence is smooth and analytic. It follows that the 1 free energy FN(β) = − β ln Z(β) is also analytic. Definition 1.2 (Phase transition). We say that a phase transition occurs, whenever the free energy density f (β) is non-analytic. Since the free entropy density φ(β) = −β f (β) is convex, the free energy density is necessarily continuous whenever it exists. The phase transitions correspond to qualitative changes in the underlying physical system.

Deﬁnition 1.3 (Types of singularities). Often, the non-analyticities occur at isolated points say βc.

• The free energy density is continuous, but its derivative with respect to β is discontinuous at βc. This singularity is called a ﬁrst-order phase transition. • The free energy and its ﬁrst derivative are continuous, but the second derivative is discontinuous at βc. This is called a second-order phase transition.

1.2 Energy spectrum and Micro-canonincal entropy density When the number of particles N grows, the volume of the configuration space increases exponentially, i.e. |XN| = |X|N. We have seen before that the system is likely to be found in lowest-energy configurations with high probability at low temperatures. From the definition of Boltzmann distribution, it is easy to check that conditioned on the system to be at certain energy level, it is equally likely to be in any configuration with equal energy. Therefore, one of the important factor of interest is the number of configurations for any given energy level. This information is given by the energy spectrum of the system, defined by the set of states with energy in the interval [E, E + ∆), N Ω∆(E) , {x ∈ X : E 6 E(x) < E + ∆}.

The number of states in Ω∆(E) is given by N∆(E) = |Ω∆(E)|.

1 Deﬁnition 1.4 (Equality in leading exponential order). We say that two quantities AN and BN are equal to leading exponential order, if 1 A lim log N = 0. N→∞ N BN . We denote this equality by AN =N BN. The energy spectrum diverges exponentially in many systems as N → ∞, if the energy is scaled linearly with N. Deﬁnition 1.5 (Micro-canonical entropy density). More precisely, there exists a function s(e) called the micro-canonical entropy density, such that given two numbers e and δ > 0,

1 0 s(e) = lim logNNδ(Ne) = sup s(e ). (2) → N ∞ N e0∈[e,e+δ] Using this notation, we can write the following equality for micro-canonical entropy density

. Ns(E/N) N∆(E) =N e . (3) The micro-canonical entropy density s(e) conveys a great amount of information about the system, and is directly related to the intensive thermodynamic potentials through a fundamental relation. Remark 1.6. Recall that energy function E : XN → R, and hence we can divide the energy levels into N∆ intervals. Then, we can partition the configuration space into configurations with energy level in one of these durations. Specifically, we can define E(x) Ω (Nk∆) = x ∈ XN : k∆ < (k + 1)∆ . N∆ 6 N

N Clearly, (ΩN∆(Nk∆) : k ∈ Z) partition the conﬁguration space X , and each of these partitions have cardi- nality |ΩN∆(Nk∆)| = NN∆(Nk∆). Therefore, we can write the partition function as   ∞ ( ) −βE(x) −βNk∆ 1 −βN( E x −k∆) ZN(β) = ∑ e = ∑ e NN∆(Nk∆)  ∑ e N . N NN∆(Nk∆) x∈X k=−∞ x∈ΩN∆(Nk∆)

. Ns(k∆) From equation (3), we get the following equality in leading exponential order NN∆(Nk∆) =N e . If X is discrete and ∆ is small enough so that the energy levels are exactly at N∆ intervals, then we can write the partition function as ∞ . N(s(k∆)−βk∆) Z(β) =N ∑ e . k=−∞ . R N(s(e)−βe) For continuous energy levels, we can show that ZN(β) =N e de, by taking limit of ∆ → 0. Proposition 1.7. If the micro-canonical entropy density (2) exists for any e and if the limit in equation (2) is uniform in e, then the free entropy density (1) exists and is given by φ(β) = max(s(e) − βe). (4) e If the maximum of s(e) − βe is unique, then the internal-energy density equals argmax(s(e) − βe). Proof. From the deﬁnition, the free entropy density φ(β) can be written as 1 φ(β) = lim log(ZN(β)). N→∞ N From the computation of N-particle partition function in Remark 1.6, we can evaluate this limit for discrete conﬁguration space as 1 ∞ φ(β) = lim exp[N(s(k∆) − βk∆)] = sup[s(k∆) − βk∆] = max[s(e) − βe]. N→∞ ∑ e N k=−∞ k

2 Example 1.8 (N identical two-level systems). We consider an N particle system, where the configuration space of each particle is identical X = {1,2} and consisting of two-levels. That is, XN = N X × X × · · · × X. For any configuration x ∈ X , we let xi ∈ X denote the configuration of ith particle. As in the previous two-level system example, we let E (x ) = 1 + 1 single i e1 {xi=1} e2 {xi=1}, where we assume e2 > e1 without any loss of generality and define the energy gap as ∆ = e2 − e1. We take the energy of the N particle system to be the sum of the single-particle energies, i.e.

E(x) = Esingle(x1) + Esingle(x2) + ... + Esingle(xN). We can next study the energy spectrum for this model. For any conﬁguration x ∈ XN, we can deﬁne the set of particles in state k ∈ {1,2} as

Sk = {i ∈ [N] : xi = k}.

We have S1 ∪ S2 = [N] and S1 ∩ S2 = ∅. If |S2| = n, then the system energy is given by

E(x) = e1(N − n) + e2n = Ne1 + n∆. N The number of possible subsets S2 ⊆ [N] such that |S2| = n is equal to the binomial coefficient ( n ). Therefore, we conclude that E(x) ∈ {E + n∆ : n = 0,. . ., N} and for any energy E = Ne1 + n∆,there are N ( n ) configurations x such that E(x) = E. This is one of the rare examples, where we can completely specify the number of configurations at each energy level, which is

E E−Ne −e N NH( n ) NH 1 NH N 1 N (E) = ≈ 2 N = 2 N∆ = 2 ∆ . ∆ n Using the deﬁnition of micro-canonical entropy density (2), we get e − e s(e) = H 1 . ∆ We can write the free energy density in terms of micro-canonical entropy density as 1 1 1 e − e f (β) = − φ(β) = − sup(s(e) − βe) = − sup H 1 − βe . β β e β e ∆ e−e1 To evaluate the supremum in the above equation, we take the ﬁrst derivative of H ∆ − βe with respect to energy density e and equate it to zero, to get e−e1 ∂H ∆ − β = 0. ∂e e=e∗ ∂H(p) Recall that ∂p = ln(1/p − 1) to obtain the stationary point exp(−β∆) e∗ = e + ∆ . 1 1 + exp(−β∆) Since H(p) is a concave function of p, it follows that e∗ corresponds to the unique maxima. Substituting this back into the expression for the free energy, we get 1 e∗ − e 1 f (β) = − H 1 − βe∗ = e − log(1 + exp(−β∆)). β ∆ 1 β This expression is identical to the free energy for a single particle as expected, since summation of energy functions amounts to no interaction system. The free energy of non-interacting N particle system, is the aggregate free energy of N independent single particle systems.